One Dimensional Melting in Single-Screw Extruders
Gregory A. Campbell, Castle Research Associates, Jonesport, ME
Mark A. Spalding, The Dow Chemical Company, Midland, MIAbstract
For special operating conditions and resins, the
melting process for single-screw extruders can occur via a
very non-traditional mechanism. For this process, melting
occurs almost exclusively at the solid bed-melt film
interface between the solid bed and the barrel wall. The
newly molten material then infiltrates the solid bed such
that a melt pool does not form on the pushing side of the
channel. This paper describes the mechanism and
provides a model for the melting process.
Introduction
Melting inside single-screw extruders typically occurs
at all four edges of the solid bed with the majority of the
melting occurring at the solid bed-melt film interface
located between the solid bed and the barrel wall. The
newly molten resin from this location is then conveyed by
the motion of the screw to a melt pool located at the
pushing side of the channel. A photograph that shows this
process via a Maddock solidification experiment is
provided in Figure 1. This melting process occurs in the
majority of single-screw extrusion operations, and it melts
the resin at relatively high rates.
Figure 1. Photograph of resin solidified in the transition section after a Maddock solidification experiment.
For very special and sometimes unpredictable
conditions, the melting process can occur by a different
mechanism. Here, the melting is occurring predominately
at solid bed-melt film interface located between the solid
bed and barrel wall. The newly molten resin infiltrates the
solid bed by flowing between the solid resin particles. A
photograph of a cross section is shown by Figure 2. This
cross section was for a specialty powder resin at a location
about 3 diameters downstream from the start of the
transition section. The pressure in the screw channels was
very low and the screw rotation rate was 30 rpm. The melting rate and extrusion rate were very low, and the
melt film between the solid bed and barrel wall was
considerably thicker than that for the conventional melting
process. The grainy cross section shown in Figure 2 is due
to a mixture of molten and solid particles at the time of the
Maddock solidification experiment. As melting progresses
in the downstream direction, the bulk temperature of the
channel increased, the level of solids decreased, and the
cross sectional view became less grainy in appearance.
Figure 2. Photograph of a specialty powder resin that is melting via the one dimensional melting mechanism. Even at this early location in the melting process, the thickness of the melt film adjacent to the barrel wall was thicker than normal.
In this paper we address the phenomena of poor melting due to one dimensional melting (or top down melting) as shown by Figure 2. This process is often
observed when extruding powders or very low viscosity
polymer pellets at conditions when the pressure in the
channel is very low. In order to understand the difference
between this melting problem and conventional melting, a
review of the development of single-screw melting theory
follows. Then the differences between the conventional
melting theory and one dimensional melting will be
developed. The model presented here compliments
previously defined models for melting in conventional
channels [1] and melting in barrier screw sections [2].
Conventional Melting Process
Experiments for studying the single-screw extrusion
melting mechanism were first carried out by Maddock [3]
and then by Street [4]. The melting mechanism was
observed by performing Maddock solidification
experiments. For these experiments, the extruder was fed
with a mixture of natural and colored (1-3%) polymer.
When steady state was achieved, screw rotation was
stopped, and full cooling was applied to the barrel,
solidifying the polymer in the channels. Usually, the
original conditions existing in the extruder were retained.
By examining the cross sections of the solidified polymer
after taking it out of the screw channel, a clear picture of
the melting behavior was obtained, as shown in Figure 1.
Their famous solidification experiments revealed that
melting in many situations takes place in a specific order.
The experiments showed that after melting began, there
was a continuous solid bed and a melt film over the inner
barrel surface. Later the solid bed was completely
surrounded by melt. Farther downstream, a melt pool
developed between the pushing flight and the solid bed.
The solid bed disappeared gradually all the way down the
channel [3].
Tadmor's research group [5,6] observed similar
melting phenomena. They studied many different
materials which included low density polyethylene
(LDPE), high density polyethylene (HDPE), acrylonitrile
butadiene styrene (ABS), rigid polyvinyl chloride (PVC),
and polypropylene (PP) resins using different operating
conditions and extruder geometries. The results were
interpreted to demonstrate that the Maddock melting
mechanism prevails in quite a large range of melting
processes with most resins. Other investigators'
experiments led to other melting phenomena, such as the
acceleration of the solid bed, break up of the solid bed,
and a melt pool existing by the trailing flight [7].
Maddock first proposed the physical description of the
melting process in 1959. In his description, the energy for
the melting process came from two sources: energy
conduction from the barrel, and viscous dissipation from
the film at the inner barrel surface. Beginning from 1966
when Tadmor [8] set up the first mathematical model
based on the Maddock melting mechanism, he proposed a
reorganizing solid bed that led to a focus on the x-dimension as rate controlling in the melting process. That
is, the rate of melting was defined as the ratio of the solid
bed width to the width of the channel.
The classical melting data originally presented by Tadmor and Klein [6] for PE resin was examined using digital dimensional analysis [10] and the melting results for three specific runs are shown Figure 3.
For essentially all of the data sets found in Tadmor
and Klein [6] the melting results resemble the data shown
in Figure 3. The data analysis is of course most difficult
near the end of melting, and in essentially all cases the
fraction of the bed remaining in the cross-channel
direction is much greater than in the thickness direction.
Also, in general the last third of the x direction material appears to melt in the last turn of the channel, creating
nearly a step function in the melting rate with respect to
the axial distance, as shown in Figure 2. Step functions, however, do not in general occur in processes that are
heat conduction dominated. To reiterate, the digital
analysis conclusively demonstrated that the y direction thickness always goes to zero before the x direction width [9,10]. This observation led us to the development
of a new physical model that is based on these new
observations. It is interesting to note that Lindt
recognized this melting behavior experimentally for
polypropylene (PP) resin in 1976 [11].
Figure 3. Melting data from Tadmor for PE resins [6,10]. At a downstream position of 25 turns, the solids were completely melted and thus the bed width and bed height were zero.
Theory Development
The new single-screw melting concept was presented
in 2009 [1] for the conventional melting process. The new
model predicted that the solid bed melted due to
dissipation and heat transfer in all four melt films
surrounding the melting solid bed, as shown in Figure 4.
Figure 4. Schematic for the zones of the new melting concept.
Consistent with the results presented in Figure 3,
separate melting dynamics were defined for the four melt
films depicted in Figure 4; C is the film between the solid
bed and the barrel wall, D is the film at the screw root (or
core) polymer interface, E is the film at the trailing flight of the screw, and B is the film (or melt pool) on the
pushing side of the screw flight.
Previous analysis of dissipation in extrusion [12] over
predicted the polymer melt temperature when barrel
rotation dissipation was used. We therefore use the screw
rotation velocities in this analysis because the process is
dissipation dominated [13-18] and dissipation is not frame
indifferent. We start with the analysis developed by
Tadmor and Klein [6]. The diagram for the new
adaptation of the Tadmor analysis is shown in Figure 5.
We start by unwrapping the screw into the x, y, and z coordinates. The velocity gradient in the z direction goes from
V
sz, the solid bed velocity in the downstream directions (
z direction), to 0 at the barrel interface. A
similar diagram was developed for the other three fluid
interfaces around the solid bed, remembering that for
these liquid interfaces the metal velocity is the screw
velocity, as shown in Figure 6.
We retain V
sy as a variable because this allows us to
calculate the change in bed thickness in our re-analysis of
the literature data, as shown in Figure 3. The
nomenclature used here follows that used by Tadmor and
Klein [6] as much as possible. Where deviations occur, a
full explanation will be presented. A nomenclature list is
presented at the end of the paper. The velocity of the solid
bed in the z direction, V
sz, is as defined by Tadmor and
Klein. The velocity of the solid bed in the
x direction, V
sx, due to screw rotation is as follows:
where N is the screw rotation speed in rpm, D is the diameter at the surface of the solid bed, theta is the helix angle at diameter D, and L is the lead length of the flight.
We calculated the thickness of the melt film, ?
c, at film C using similar assumptions used by Tadmor and
Klein [6] and thus solve their Equation (5-23) for V
sy,C, the velocity of the loss of the solid bed due to melting at ?c.
where k
m is the thermal conductivity of the melt, T
b is the barrel temperature, T
m is the melting temperature of the resin, micro is the shear viscosity, V
j is the vectorial velocity in the film (discussed later), ρ
s is the density of the compacted solid bed, lamda is the heat of fusion, c
s is the heat capacity, and T
s is the temperature of the solid bed. The term V
sy,C here is equivalent to Tadmor and Klein’s V
sy. Equation (3) is simply an energy balance for the melting process [1]. Melting velocities at the other three interfaces V
sy,D, V
sx,B, and V
sx,E are essentially identical to that for Equation (3) except that the vectorial velocity used corresponds to the specific film. These equations were omitted here.
Figure 5. Melting differential element for a rotating screw for the melt film in zone C.
The melting velocities of the solid bed interfaces are
the foundation of this new approach. The mathematical
equality to calculate the rate of loss of the solid at the
interface next to the barrel is found in Equation (4).
where Y is the local thickness of the solid bed and t is the independent time variable.
The melt film thickness for zone C, ?, is determined using Equation (5). Note that V
sx is essentially numerically
equal to the traditional V
bx as specified by Tadmor and Klein because of the small value of the film thickness.
where X is the local width of the solid bed and α is a
fitting factor. Equation (5) is a mass balance for the zone
C film and is essentially identical to Tadmor and Klein’s
[6] Equation (5-31). The flow equation is a first-order
approximation since it neglects flows due to pressure
gradients. Ideally, the value of α is set to 2 so as to
provide an average thickness for the film in the crosschannel
direction. The value of α was set to 0.5 for the
simulation here to force the melting lengths to be
equivalent to those in the literature.
The film thickness, ?
e, for zone E was determined
using a similar mass balance equation as follows:
where Z is downstream position of the solid bed. The melt film thicknesses, ?
b and ?
d for zones B and D were found by difference:
where W is the width of screw channel and H is the local
depth of channel.
The boundary conditions for the velocity at the screw
root interfaces are shown in Figure 6.
The results of this analysis can be found in a previous
paper [1] and are referred to here as the new melting
model.
One-Dimensional Melting Model
The problem at hand has different dissipation physics.
In this case the solid bed is conceptualized as depicted in
Figure 7.
Comparing Figure 7 to Figure 4 it is observed that
melting zones B, D, and E are no longer available for
producing dissipation to aid the melting. There is some
free space around the solid which allowed the solid to be
pushed up the transition. However this space is not wide
enough to allow the fluid at the top of the bed to
encapsulate the bed. Thus the expressions for
?
D, ?
B, and ?
E now reduce to zero. Also the thickness of the film in C,
where we have previously demonstrated that it produces
the greatest melting flux [1], will now not just be modeled
as the Tadmor model for ?
c via Equation 5. In this
analysis the initial film thickness is defined the same
as for our new melting model; i.e., Equation 5. Then
as the solid bed melts at the barrel interface the ?
c becomes a function of the channel thickness and the solid
bed remaining.
Figure 6. Melting differential element at the screw surfaces for a rotating screw reference frame.
Figure 7. Conceptual melting physics for one dimensional melting.
where the film thickness in now ?
C_melt once the melting starts. Here H
0 is the initial channel thickness, A
z is the rate that the channel decreases in depth with the down
channel distance Z, and Y(Z) is the bed thickness at any Z.
This increase in film thickness at the barrel solid bed
interface decreases the dissipation because the local shear
rate decrease as the film thickness increases. It will be
found that this caused the bed to melt much more slowly.
Results and Discussion¶
We are now in a position to determine if the proposed
theory will produce results that are consistent with
reported observations that these types of materials melt
more slowly because the bed has not been encapsulated.
This new theory requires the solution of only two
coupled differential equations, unlike the model presented
in 2009 where there are five simultaneously coupled
equations. Here x, y, and z are the local coordinates of the
solid bed:
We used a differential equation solver to obtain the
results of the initial value problem.
Typical simulation results are presented in Figure 8
for the conventional melting model with melting in all
four melt films, C, D, E, and B. As shown by this figure,
the Y thickness goes to zero at a down channel position of
220 cm into the melting section. It is seen that the change
in the thickness (x direction) leaves about 60% of the
original bed width remaining just at the end of melting.
The end of melting is defined as when the y direction
thickness of the solid bed goes to zero and thus the x
direction thickness also goes to zero because there is no
more material to melt. This simulation was performed
using a PE resin with a shear viscosity of 880 Pa-s.
Using the same PE resin with the same physical
properties, the model was modified to represent the
melting physics shown in Figure 7. The results are
presented in Figure 9. It is observed that the bed width
remains constant because there is no dissipation at the
screw helix wall, zones B and E in the new model physics,
as shown in Figure 4. For the one dimensional melting
from the barrel surface, unlike the results in Figure 8
where the bed thickness was consumed at a down channel position of about 220 cm, about 25% of the original bed
remains.
Figure 8. Simulation results using the new melting model [1] for the conventional melting process; i.e., melting is occurring in all four melt films.
Figure 9. Simulation results for melting only from the barrel surface; i.e., one dimensional melting. About 75% of the resin was melted at end of the transition section.
One dimensional melting and melt infiltration into the
solid bed will reduce the melting capacity of the process,
as shown by Figure 9. The melting mechanism can switch
from the normal conventional melting to the poor melting
performance shown above due to the lack of pressure in
the melting channel. Pressure in the channel is needed to
compact the solid polymer feedstock [19] into a
nonporous bed, eliminating the possibility of melt
infiltration. Melting rates are also known to increase with
increasing pressure in the channel [20]. Moreover, as the
melt film between the barrel and solid bed becomes
thicker in the downstream direction, energy dissipation
and the melting rate will be reduced. With reduced energy
dissipation, the effect of energy conduction becomes more
important.
Conclusions
A new approach has been developed that correlates
with observations of several polymer systems that do not melt in the conventional manner. The new alternative
model proposed for the melting of the solid polymer bed
in a single-screw extruder has been modified to analyze
the melting phenomena of a solid bed in a single location
and allows the infiltration of molten resin into a porous
solid bed. The model alters the heat transfer and melting
velocities proposed in our previous paper where a new
melting concept was modeled [10]. This modeling effort
demonstrates that as the melt film increases at the barrel
solid interface the dissipation decreased due to lower
shear rate and the bed melts much slower.
Nomenclature
c
s heat capacity of the solid.
D diameter at the surface of the solid bed.
H local channel depth.
k
m thermal conductivity of the melt.
L lead length of the flight.
N screw rotation speed in rpm.
t independent time variable.
T
b barrel temperature.
T
m melting temperature of the resin.
T
s temperature of the solid bed.
V
j vectorial velocity in the films.
V
bx velocity of the barrel in the x direction for a barrel rotation reference frame (not used).
V
sx velocity of the solid bed in the x direction.
V
sy velocity of the solid bed consumption in the thickness direction at the zone C film (Tadmor).
V
sy,C velocity of the solid bed consumption in the thickness direction at the zone C interface (y direction).
V
sz velocity of the solid bed in the z direction.
W width of the channel perpendicular to the flight.
X width of the solid bed.
x independent variable in the cross-channel direction.
Y thickness of the solid bed in the y direction.
y independent variable in the channel-depth direction.
Z downstream position of the solid bed.
z independent variable in the down-channel direction.
α fitting factor.
theta helix angle at diameter D.
ρ
m density of the molten polymer.
ρ
s density of the compacted solid bed.
λ heat of fusion.
?
b thickness of melt film B.
?
c thickness of melt film C.
?
c_melt thickness of melt film C 1 D melting.
?
d thickness of melt film D.
?
e thickness of melt film E.
μ shear viscosity.
Acknowledgments
We would like to thank The Dow Chemical
Company, Becton Dickinson, Eastman Chemical,
XALOY, American Leistritz, 3M, and Pechiney. Much of
this work was initiated in the Polymer Extrusion
Consortium at Clarkson University.
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Key Words: Extrusion, Melting, Single-Screw,
Mechanism, Theory, Modeling.
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